Math Behind PMM

Token Type

base and quote are two concepts that will be mentioned frequently. Two easy ways to distinguish between them are:

• In a trading pair, the base is always the token before the hyphen, and quote is after it

• In transactions, the price refers to how many quote tokens are needed in exchange for one base token

For example, in the ETH-USDC trading pair, ETH is the base token and USDC is the quote token.

Pool Parameters

The funding pool of PMM is described by four parameters:

• $B_0$: base token regression target - total number of base tokens deposited by liquidity providers
• $Q_0$: quote token regression target - total number of quote tokens deposited by liquidity providers
• $B$: base token balance - number of base tokens currently in the pool
• $Q$: quote token balance - number of quote tokens currently in the pool

PMM Formula

The PMM price curve is plotted by the following pricing formula:

$P_{margin}=iR$

Where $R$ is defined to be the piecewise function below:

$SellBaseToken：if \ B<B_0, \ R=1-k+(\frac{B_0}{B})^2k$

$Sell Quote Token：if \ Q<Q_0, \ R=1/(1-k+(\frac{Q_0}{Q})^2k)$

$else \ R=1$

AMM & Stable Swap Use Cases

PMM supports transition to AMM and stable coin trading as well.

Traditional AMM

PMM is a generalization of AMM, when:

• k = 1

• deposit and withdraw in proportion to the current pool ratio

Stable swap

PMM can also be used as stable swap, when:

• i = 1

• k = 0.0001